Expand Using the Binomial Theorem (1+x/4)^8

${(1 + \frac{x}{4})}^{8}$

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{8} \frac{8!}{(8 - k)! k!} \cdot {(1)}^{8 - k} \cdot {(\frac{x}{4})}^{k}$

Expand the summation.

$\frac{8!}{(8 - 0)! 0!} \cdot {(1)}^{8 - 0} \cdot {(\frac{x}{4})}^{0} + \frac{8!}{(8 - 1)! 1!} \cdot {(1)}^{8 - 1} \cdot {(\frac{x}{4})}^{1} + \dots + \frac{8!}{(8 - 8)! 8!} \cdot {(1)}^{8 - 8} \cdot {(\frac{x}{4})}^{8}$

Simplify the exponents for each term of the expansion.

$1 \cdot {(1)}^{8} \cdot {(\frac{x}{4})}^{0} + 8 \cdot {(1)}^{7} \cdot {(\frac{x}{4})}^{1} + \dots + 1 \cdot {(1)}^{0} \cdot {(\frac{x}{4})}^{8}$

Simplify the polynomial result.

$1 + 2 x + \frac{7 x^{2}}{4} + \frac{7 x^{3}}{8} + \frac{35 x^{4}}{128} + \frac{7 x^{5}}{128} + \frac{7 x^{6}}{1024} + \frac{x^{7}}{2048} + \frac{x^{8}}{65536}$

Expand Using the Binomial Theorem (1+x/4)^8

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